How Does the Dot Product of Two Vectors Work?

Date: 03/17/2006 at 08:15:50
From: Sheri
Subject: dot products and identity functions

I'm a senior in high school studying advanced math and we are now 
studying about the dot product.  My question is, how do we describe 
the identity function in terms of the dot product formula?  Our 
teacher was talking about how they're related but I'm absolutely 
confused!  I don't even know what the dot product is about and how 
does multiplying two vectors give you the angle they form together?

Date: 03/17/2006 at 09:05:15
From: Doctor Jerry
Subject: Re: dot products and identity functions

Hello Sheri,

Thanks for writing to Dr. Math.

I'll be referring to this image:

   

The vectors u and v are given, t is the angle between them (typing "t" 
is easier than typing "theta"), and w is the length of the line 
joining the tips of u and v.  The components of u and v are u1, u2, 
v1, and v2.  You may usually write u = u1*i + u2*j.

The length of w can be calculated using the law of cosines.  For this,
notice that the coordinates of the tips of u and v are (u1,u2) and
(v1,v2).  From the law of cosines and letting sqrt(u1^2 + u2^2) = |u|,
the length of u, and sqrt(v1^2 + v2^2) = |v|, the length of v, we see
that:

  w^2 = |u|^2 + |v|^2 - 2*|u|*|v|*cos(t)

Because w^2 = (u1-v1)^2 + (u2-v2)^2, we see that:

  (u1-v1)^2 + (u2-v2)^2 =  |u|^2 + |v|^2 - 2*|u|*|v|*cos(t)

If we now replace |u|^2 and |v|^2 by u1^2 + u2^2 and v1^2 + v2^2 we find:

  (u1-v1)^2 + (u2-v2)^2 = u1^2 + u2^2 + v1^2 + v2^2 - 2*|u|*|v|*cos(t)

Expanding the left side and simplifying:

  u1*v1 + u2*v2 = |u|*|v|*cos(t)

The left side is usually given the name "dot product" and written as
"u.v", so:

  u.v = |u|*|v|*cos(t)

As I hope you find out, this is a very useful result.

Please write back if my comments are not clear.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/